I made this by augmentation (with prisms) of the faces of a small stellated dodecahedron, using *Stella 4d*, which you can try for free at this website.

# Monthly Archives: February 2020

# A Symmetrohedron Featuring Twelve Regular Pentagons, Thirty Regular Octagons, and Twenty Equilateral Triangles

In addition to the sixty-two regular faces, this polyhedron also has two sets of sixty isosceles trapezoids each, shown in different shades of blue. That’s 182 faces in all. I made it using *Stella 4d*, which you can try for free at www.software3d.com/Stella.php.

# A Partial Faceting of the Rhombic Octagonoid

The rhombic octagonoid appears here. These polyhedra were made using *Stella 4d*, which you can try for free at this website.

# The Rhombic Octagonoid, a Zonohedron With Ninety Faces

To make this zonohedron with *Stella 4d* (available as a free trial download here), start with a dodecahedron, and then perform a zonohedrification based on both faces and vertices. It is similar to the rhombic enneacontahedron, with thirty equilateral octagons replacing the thirty narrow rhombic faces of that polyhedron.

I’ve run into this polyhedron from time to time, and I’ve also had students make it. It is the largest zonohedron which can be built using only red and yellow Zome (available here). I thought it needed a name, so I made one up.

# A 122-Faced Symmetrohedron Featuring Twelve Regular Decagons and Thirty Regular Octagons

I previously blogged a different version of this symmetrohedron, one which sacrifices octagon regularity for regularity of all the other faces. Both polyhedra were created using *Stella 4d: Polyhedron Navigator,* which you can try for free right here. In this version, the twenty red hexagons are equiangular, and the sixty green faces are isosceles trapezoids.

# From the Rhombic Enneacontahedron to an All-Kite Polyhedron

This is the rhombic enneacontahedron, one of the few well-known zonohedra. Its ninety faces have two types: sixty wide rhombi, and thirty narrow rhombi.

In the image above, the thirty narrow rhombi of the rhombic enneacontahedron have been augmented with prisms

The next step in my polyhedral play was to create the convex hull of this augmented rhombic enneacontahedron. This produced the solid shown above. To make the one shown below, I next used a function called “try to make faces regular.” The result is a symmetrohedron with 122 faces: 12 regular pentagons, 30 rhombi, 60 almost-square isosceles trapezoids, and thirty equilateral triangles.

Finally, I examined the dual of this symmetrohedron, which turned out to have 120 faces: two sets of sixty kites each.

The program I used to create these polyhedral images is called *Stella 4d*, and you can try it yourself (as a free trial download) at http://www.software3d.com/Stella.php.

# A Pyramid-Augmented Rhombicosidodecahedron

All of the edges of this polyhedron have the same length. I made it using *Stella 4d: Polyhedron Navigator*, which you can try for free at this website.